On the Rate of Convergence of Empirical Barycentres in Metric

On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics Adil Ahidar-Coutrix, Thibaut Le Gouic, Quentin Paris

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Adil Ahidar-Coutrix, Thibaut Le Gouic, Quentin Paris. On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics. 2019. hal-01810530v3

HAL Id: hal-01810530 https://hal.archives-ouvertes.fr/hal-01810530v3 Preprint submitted on 17 Jun 2019

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Given a separable and complete metric space (M, d), deﬁne (M) as the set of Borel probability measures P2 P on M such that d(x,y)2 dP (y) < + , ∞ ZM for all x M. A barycenter of P (M), also called a Fréchet mean [Fré48], is any element x∗ M ∈ ∈ P2 ∈ such that x∗ arg min d(x,y)2 dP (y). (1.1) ∈ x∈M ZM When it exists, a barycenter stands as a natural analog of the mean of a (square integrable) probability measure on Rd. Alternative notions of mean value include local minimisers [Kar14], p-means [Yok17], exponential barycenters [ÉM91] or convex means [ÉM91]. Extending the notion of mean value to the case of probability measures on spaces M with no Euclidean (or Hilbert) structure has a number of applications ranging from geometry [Stu03] and optimal transport [Vil03, Vil08, San15, CP19] to statistics and data science [Pel05, BLL15, BGKL18, KSS19], and the context of abstract metric spaces provides a unifying framework encompassing many non-standard settings.

Properties of barycenters, such as existence and uniqueness, happen to be closely related to geometric characteristics of the space M. These properties are addressed in the context of Riemannian manifolds in [Afs11]. Many interesting examples of metric spaces, however, cannot be described as smooth manifolds because of their singularities or inﬁnite dimensional nature. More general geometrical structures are geodesic metric spaces which include many more examples of interest (precise deﬁnitions and necessary background on metric geometry are reported in Appendix A). The barycenter problem has been addressed in this general setting. The scenario where M has non-positive curvature (from here on, curvature bounds are understood in the sense of Aleksandrov) is considered in [Stu03]. More generally, the case of metric spaces with upper bounded curvature is studied in [Yok16] and [Yok17]. The context of spaces M with lower bounded curvature is discussed in [Yok12] and [Oht12].

Focus on the case of metric spaces with non-negative curvature may be motivated by the increasing interest for the theory of optimal transport and its applications. Indeed, a space of central importance in this context is the Wasserstein space M = (Rd), equipped with the Wasserstein metric W , known P2 2 to be geodesic and with non-negative curvature (see Section 7.3 in [AGS08]). In this framework, the barycenter problem was first studied by [AC11] and has since gained considerable momentum. Existence and uniqueness of barycenters in (Rd) has further been studied in [LL17]. P2 A number of objects of interest, including barycenters as a special case, may be described as minimisers of the form x∗ arg min F (x,y) dP (y), (1.2) ∈ x∈M ZM for some probability measure P on metric space M and some functional F : M M R. While we × → obviously recover the definition of barycenters whenever F (x,y) = d(x,y)2, many functionals of interest are not of this specific form. With a slight abuse of language, minimisers such as x∗ will be called generalised barycenters in the sequel. A first example we have in mind, in the context where M = (Rd), P2 is the case where functional F is an f-divergence, i.e.

f dµ dν if µ ν, F (µ,ν) := dν ≪ ( + otherwise, R ∞ for some convex function f : R R. Known for their importance in statistics [Le 86, Tsy09], and + → information theory [Vaj89], f-divergences have become a crucial tool in a number of other ﬁelds such as

2 geometry and optimal transport [Stu06a, Stu06b, LV09] or machine learning [GPAM+14]. Other examples arise when the squared distance d(x,y)2 in (1.1) is replaced by a regularised version F (x,y) aiming at enforcing computationally friendly properties, such as convexity, while providing at the same time a sound approximation of d(x,y)2. A signiﬁcant example in this spirit is the case where functional F is the entropy-regularised Wasserstein distance (also known as the Sinkhorn divergence) largely used as a proxy for W2 in applications [Cut13, CP19, AWR17, DGK18].

In the paper, our main concern is to provide rates of convergence for empirical generalised barycenters, deﬁned as follows. Given a collection Y1,...,Yn of independent and M-valued random variables with same distribution P , we call empirical generalised barycenter any

n 1 xn arg min F (x,Yi). (1.3) ∈ x∈M n Xi=1 ∗ Any such xn provides a natural empirical counterpart of a generalised barycenter x defined in (1.2). The 2 statistical properties of xn have been studied in a few specific scenarios. In the case where F (x,y)= d(x,y) and M is a Riemannian manifold, significant contributions, establishing in particular consistency and limit distribution under general conditions, are [BP03, BP05] and [KL11]. Asymptotic properties of empirical barycenters in the Wasserstein space are studied in [LL17]. We are only aware of a few contributions providing finite sample bounds on the statistical performance of xn. Paper [BGKL18] provides upper and lower bounds on convergence rates for empirical barycenters in the context of the Wasserstein space over the real line. Independently of the present contribution, [Sch18] studies a similar problem and provides results complementary to ours.

In addition to more transparent conditions, our results are based on the fundamental assumption that there exists constants K > 0 and β (0, 1] such that, for all x M, ∈ ∈ β d(x,x∗)2 K (F (x,y) F (x∗,y)) dP (y) . (1.4) ≤ − ZM We show that condition (1.4) provides a connection between usual assumptions in the field of empirical processes and geometric characteristics of the metric space M. First, the reader familiar with the theory of empirical processes will identify in the proof of Theorems 2.1 and 2.5 that condition (1.4) implies a Bernstein condition on the class of functions indexing our empirical process, that is an equivalence between their L2 and L1 norms. Many authors have emphasised the role of this condition for obtaining fast rates of convergence of empirical minimisers. Major contributions in that direction are for instance [MT99, Mas00, BLV03, BBM05, BJM06, Kol06, BM06] and [Men15]. In particular, this assumption may be understood in our context as an analog of the Mammen-Tsybakov low-noise assumption [MT99] used in binary classification. Second, we show that condition (1.4) carries a strong geometrical meaning. In the context where F (x,y) = d(x,y)2, [Stu03] established a tight connection between (1.4), with K = β = 1, and the fact that M has non-positive curvature. When F (x,y) = d(x,y)2, we show that (1.4) actually holds with K > 0 and β = 1 in geodesic spaces of non-negative curvature under flexible conditions related to the possibility of extending geodesics emanating from a barycenter. Finally, for a general functional F , we connect (1.4) to its strong convexity properties. Using terminology introduced in [Stu03] in a slightly more specific context, we will call by extention (1.4) a variance inequality.

The paper is organised has follows. Section 2 provides convergence rates for generalised empirical barycenters under several assumptions of functional F and two possible complexity assumptions on metric space M. Section 3 investigates in details the validity of the variance inequality (1.4) in diﬀerent scenarios. In particular, we focus on studying (1.4) under curvature bounds of the metric M whenever F (x,y) = d(x,y)2. Additional examples where our results apply are discussed in Section 4. Proofs are postponed to

3 Section 5. Finally, Appendix A presents an overview of basic concepts and results in metric geometry for convenience.

2 Rates of convergence

In this section, we provide convergence rates for generalised empirical barycenters. Paragraph 2.1 deﬁnes our general setup and mentions our main assumptions on functional F . Paragraphs 2.2 and 2.3 present rates under diﬀerent assumptions on the complexity of metric space M. Subsection 2.4 discusses the optimality of our results.

2.1 Setup Let (M, d) be a separable and complete metric space and F : M M R a measurable function. Let × → P be a Borel probability measure on M. Suppose that, for all x M, the function y M F (x,y) is ∈ ∈ 7→ integrable with respect to P , and let

x∗ arg min F (x,y) dP (y), (2.1) ∈ x∈M ZM which we suppose exists. Given a collection Y1,...,Yn of independent and M-valued random variables with same distribution P , we consider an empirical minimiser

n 1 xn arg min F (x,Yi). (2.2) ∈ x∈M n Xi=1 The present section studies the statistical performance of xn under the following assumptions on F .

(A1) There exists a constant K > 0 such that, for all x,y M, 1 ∈ F (x,y) K . | |≤ 1 (A2) There exist constants K > 0 and α (0, 1] such that, for all x,x′,y M, 2 ∈ ∈ F (x,y) F (x′,y) K d(x,x′)α. | − |≤ 2 (A3) (Variance inequality) There exist constants K > 0 and β (0, 1] such that, for all x M, 3 ∈ ∈ β d(x,x∗)2 K (F (x,y) F (x∗,y)) dP (y) . ≤ 3 − ZM Assumptions (A1) and (A2) are transparent boundedness and regularity conditions. For instance 2 2 if F (x,y) = d(x,y) , these assumptions are satisﬁed whenever M is bounded with K1 = diam(M) , K2 = 2diam(M) and α = 1, by the triangular inequality. The meaning of condition (A3) is less obvious at ﬁrst sight. A detailed discussion of (A3) is postponed to section 3. For now, we mention three straightforward implications of (A3). First, note that imposing both (A2) and (A3) requires M to be bounded. Indeed, plugging (A2) into (A3) yields

β 1 diam(M) 2(K K ) 2−αβ . ≤ 2 3 More importantly, (A3) implies that minimiser x∗ is unique. Finally, condition (A3) applied to minimiser xn reads β d(x ,x∗)2 K (F (x ,y) F (x∗,y)) dP (y) . (2.3) n ≤ 3 n − ZM 4 The left hand side of this inequality is the estimation performance of xn. The integral, under power β, may be called the learning performance of xn, or its excess risk. Having this comparison in mind, we will ∗ 2 focus on controlling the learning performance of xn knowing that an upper bound on d(xn,x ) may be readily deduced from our results. The remainder of the section therefore presents upper bounds for the right hand side of (2.3) under specific complexity assumptions on M. For that purpose, we recall the definition of covering numbers. For A M and ε> 0, an ε-net for A is a finite subset x ,...,x M ⊂ { 1 N }⊂ such that N A B(x , ε), ⊂ j j[=1 where B(x, ε) := u M : d(x, u) < ε . The ε-covering number N(A, d, ε) (1, + ] is the smallest { ∈ } ∈ ∞ integer N 1 such that there exists an ε-net of size N for A in M. The function ε log N(A, d, ε) will ≥ 7→ be referred to as the metric entropy of A.

2.2 Doubling condition Our ﬁrst complexity assumption is the following.

(B1) (Doubling condition) There exist constants C,D > 0 such that, for all 0 < ε r, ≤ Cr D N(B(x∗, r), d, ε) . ≤ ε Condition (B1) essentially characterises M as a D-dimensional space and implies the following result.

Theorem 2.1. Assume that (A1), (A2), (A3) and (B1) hold. Then, for all n 1 and all t> 0, ≥ 1 1 2 D 2−αβ t 2−αβ kd(x∗,x ) β (F (x ,y) F (x∗,y)) dP (y) A max , , n ≤ n − ≤ · n n ZM ( )

− 1 with probability at least 1 2e−t, where k = K β and A is an explicit constant independent of n. − 3 Note that bounds in expectation may be derived from this result, using classical arguments. As described in section 3, (A2) and (A3) hold in several interesting cases for α = β = 1. In this case, Theorem 2.1 exhibits an upper bound of order D/n. A discussion on the optimality of this result is postponed to paragraph 2.4 below. Next, we shortly comment condition (B1).

Remark 2.2. With a slight abuse of terminology, condition (B1) is termed doubling condition. In the literature, the doubling condition usually refers to the situation where inequality

sup sup log2 N(B(x, 2ε), d, ε) D x∈M ε>0 ≤ holds for some D (0, + ). It may be seen that this inequality implies (B1) with C = 2. Note however ∈ ∞ that (B1) is slightly less restrictive as it requires only the control of the covering numbers of balls centered at x∗. This fact is sometimes useful as described in example 2.4 below.

We now give two examples where assumption (B1) holds.

Example 2.3. Suppose there exists a positive Borel measure µ on (M, d) such that, for some D > 0, we have D D (x, r) M (0, + ), α−r µ(B(x, r)) α r , (2.4) ∀ ∈ × ∞ ≤ ≤ +

5 for some constants 0 < α− α < + . Then, for all x M and all 0 < ε r, ≤ + ∞ ∈ ≤ α r D α 3r D − N(B(x, r), d, ε) + , (2.5) α+ ε ≤ ≤ α− ε and thus (B1) is satisﬁed. The proof is given in Section 5. Measures µ satisfying condition (2.4) are called D-regular or Ahlfors-David regular. Many examples of such spaces are discussed in section 12 in [GL00] or section 2.2 in [AT04]. Note that the present example includes the case where M is a D-dimensional compact Riemannian manifold equipped with the volume measure µ. A direct and simple consequence of Example 2.3 is that (B1) holds in any D-dimensional vector space equipped with any norm since the Lebesgue measure satisﬁes (2.4) with α− = α+. While simple in essence, this observation allows to exhibit more general parametric families satisfying (B1) as in the next example. Example 2.4 (Location-scatter family). Here, we detail an example of a subset M of the Wasserstein space ( (Rd),W ) for which assumption (B1) holds. We say that M (Rd) is a location-scatter family P2 2 ⊂ P2 if the following two requirements hold: (1) All elements of M have a non-singular covariance matrix. (2) For every two measures µ ,µ M, with expectations m and m and with covariance matrices Σ 0 1 ∈ 0 1 0 and Σ1 respectively, the map 1/2 T : x (m m )+Σ−1/2 Σ1/2Σ Σ1/2 Σ−1/2x 01 7→ 1 − 0 0 0 1 0 0 pushes forward µ to µ , i.e. µ (T −1(A)) = µ (A) for any Borel set A Rd, which we denote 0 1 0 01 1 ⊂ (T01)#µ0 = µ1.

Such sets have been studied for instance in [ÁEdBCAM16]. The map T01 being the gradient of a convex function, the theory of optimal transport guarantees that the coupling (id, T01)#µ0 is optimal, so that

W 2(µ ,µ )= x T (x) 2dµ (x) 2 0 1 k − 01 k 0 Z 1/2 = m m 2 + tr Σ +Σ 2 Σ1/2Σ Σ1/2 , (2.6) k 0 − 1k 0 1 − 0 1 0 where tr(A) denotes the trace of matrix A and . refers to the standard euclidean norm. Next, we show k k that such a family satisﬁes (B1). Let P be a probability measure on M and denote µ∗ M a barycenter ∈ of P with mean m∗ and covariance matrix Σ∗. For any two measures µ ,µ M, set 0 1 ∈ −1/2 1/2 1/2 1/2 −1/2 T∗ (x)= m m∗ +Σ∗ Σ∗ Σ Σ∗ Σ∗ x, i i − i ∗ where mi and Σi denote the mean and covariance matrix of µi, i = 0, 1. The pushforward (T∗0, T∗1)#µ is a (possibly suboptimal) coupling between µ0 and µ1. Therefore,

2 2 ∗ W (µ ,µ ) T∗ (x) T∗ (x) dµ (x) (2.7) 2 0 1 ≤ k 0 − 1 k Z 1/2 1/2 2 −1/2 1/2 1/2 −1/2 1/2 1/2 2 = m m + Σ∗ Σ∗ Σ Σ∗ Σ∗ Σ∗ Σ Σ∗ , k 0 − 1k k 0 − 1 kF 2 2 2 where . stands for the Frobenius norm. Note that (m, A) = m + A deﬁnes a norm . ∗ on k kF k k∗ k k k kF k k the vector space Rd S where S denotes the space of symmetric matrices of size d d. Then, deﬁne the × d d × function φ that maps each µm,Σ in the location-scatter family, with mean m and covariance Σ, to

−1/2 1/2 1/2 1/2 φ(µm,Σ)= m, Σ∗ Σ∗ ΣΣ∗ . 6 Then, combining (2.6) and (2.7), it follows that

W (µ ,µ ) φ(µ ) φ(µ ) ∗, (2.8) 2 0 1 ≤ k 0 − 1 k ∗ ∗ with equality if µ0 = µ or µ1 = µ . Therefore, since φ(M) is a subset of a vector space of dimension D = d + d(d + 1)/2, there exists C > 0 such that for all ε> 0,

D ∗ 1/2 Cr N(B(µ , r),W , ε) N(B((m∗, Σ∗ ), r) φ(M), . ∗, ε) . 2 ≤ ∩ k k ≤ ε Hence, (B1) holds. The result, derived in example 2.4, may be generalised to other parametric subsets of the Wasserstein space (or more generally parametric subsets of geodesic Polish spaces with non-negative curvature). Indeed, since the Wasserstein space over Rd has non-negative curvature, the support of P pushed forward to the tangent cone at a barycenter µ∗ is isometric to a Hilbert space (this result follows by combining Theorem 5.5 and Lemma 5.8) and its norm satisfies (2.8) (see Proposition A.13). Therefore it is enough, for (B1) to hold, to require that the image of the support of P by the logµ∗ map (see paragraph A.5 for a definition) is included in a finite dimensional vector space.

2.3 Polynomial metric entropy Condition (B1) is essentially a ﬁnite dimensional behaviour and does not apply in some scenarios of interest. This paragraph addresses the situation where the complexity of set M, measured by its metric entropy, is polynomial. (B2) (Polynomial metric entropy) There exists constants C,D > 0 such that, for all ε> 0,

C D log N(M, d, ε) . ≤ ε Theorem 2.5. Assume that (A1), (A2), (A3) and (B2) hold. Then, for all n 1 and all t> 0, ≥ 1 2 t 2−αβ kd(x∗,x ) β (F (x ,y) F (x∗,y)) dP (y) A max v , , n ≤ n − ≤ · n n ZM ( ) with probability at least 1 2e−t, where − − 2 n 4−(2α−D)β if D< 2α, vn =  (log n)/√n if D = 2α, − α  n D if D> 2α,

− 1  β  where k = K3 and A is an explicit constant independent on n. As for Theorem 2.1, bounds in expectation may be easily derived from this result. The optimality of Theorem 2.5 is addressed in paragraph 2.4. Next is an example where assumption (B2) applies. Example 2.6 (Wasserstein space). Let B = x Rd : x x ρ be a closed Euclidean ball in Rd { ∈ k − 0k2 ≤ } and let M = (B) be the set of square-integrable probability measures supported on B equipped with the P2 2-Wassertein metric W2. Combining the result of Appendix A in [BGV07] with a classical bound on the covering number of euclidean balls, it follows that for all 0 < ε ρ, ≤ 6ρ d 8eρ log N(M,W , ε) 2 log . 2 ≤ ε ε 7 In particular, for any D > d, there exists C > 0 depending on D and ρ such that, for all 0 < ε ρ, ≤ C D log N(M,W , ε) , 2 ≤ ε so that (B2) is satisﬁed for all D > d. We ﬁnally point towards Theorem 3 in [WB19] which may be used to derive upper bounds on the covering number of subsets of the 2-Wasserstein space composed of measures, absolutely continuous with respect to the Lebesgue measure, and with density belonging to some Besov class.

2.4 On optimality At the level of generality considered by Theorems 2.1 and 2.5, we have not been able to assess the optimality of the given rates for all choices of functional F satisfying the required assumptions and all values of constants D, α, β. In particular, it is likely that the rates displayed in Theorem 2.5 are artefacts of our proof techniques and that results may be improved in some speciﬁc scenarios using additional information on the problem at hand. However, we discuss below some regimes where our results appear sharp and, on the contrary, settings where our results should allow for improvements.

To start our discussion, consider the barycenter problem, i.e. the case where F (x,y)= d(x,y)2. In the context where (M, d) is a Hilbert space equipped with its usual metric and P is square integrable, explicit n computations reveal that xn = i=1 Yi/n is an empirical barycenter of P in the sense of (2.2) and that x∗ = E[Y ] (in the sense of the Pettis or Bochner integral) is the unique barycenter of P . In addition, we 1 P check that, for all n 1, ≥ 1 Ed(x ,x∗)2 = E (d(x ,x)2 d(x∗,x)2)dP (x)= d(x,x∗)2 dP (x). (2.9) n n − n ZM ZM We notice that, under assumptions much more general than those considered in the present paper, the rate of convergence (in expectation) of empirical barycenters in a Hilbert space is of order 1/n. While this observation concerns the very special case of Hilbert spaces, we conjecture that the rates of convergence of empirical barycenters is of order 1/n in a wide family of metric spaces including Hilbert spaces as a special case. Identifying precisely this wider family remains an open question but it appears from this discussion that boundedness and complexity restrictions, such as (A1), (B1) and (B2), may be unnecessary for the barycenter problem. Whenever (M, d) is RD equipped with a general norm, a very interesting recent contribution, connected to that question, is [LM19]. On a more positive note, we point towards two encouraging aspects encoded in our results in the context of the barycenter problem. First, consider the D case where (M, d) is R equipped with the euclidean metric and suppose that the Yi’s are independent with gaussian distribution (x∗,σ2). Then, identity (2.9) reads in this case N σ2D Ed(x ,x∗)2 = E (d(x ,x)2 d(x∗,x)2)dP (x)= . n n − n ZM It is known, furthermore, that σ2D/n corresponds (up to universal constants) to the minimax rate of ∗ ∗ estimation of x in the context where the Yi’s are i.i.d. subgaussian random variables with mean x and variance proxy σ2 (see Chapter 4 in [RH17]). Therefore, provided (M, d) is a bounded metric space (which guarantees (A1) and (A2) with α = 1) and provided assumption (A3) holds for β = 1 (which is often the case as discussed in paragraphs 3.1 and 3.2 below) Theorem 2.1 recovers the optimal rate of convergence D/n, up to constants, in a fairly wide context. Finally, note that while possibly suboptimal in some cases, the rates provided by Theorems 2.1 and 2.5, combined with examples 2.4, 2.6 and discussions of paragraph 3.2, provide up to our knowledge the ﬁrst rates for the Wasserstein barycenter problem at this level of generality. An exception is the Wasserstein space over the real line (studied, for instance, in [BGKL18])

8 which happens to be isometric to a convex subset of a Hilbert space as can be deduced for instance from combining statement (iii) of Proposition 3.5 in [Stu03] and Proposition 4.1 in [Klo10].

Outside from the setting of the barycenter problem, not much is known on optimal rates of estimation or learning (in the sense described at the end of paragraph 2.1) of x∗ deﬁned in (2.1). We believe this question remains mainly open. It is our impression that the 1/n rate, conjectured to hold for empirical barycenters in a wide setup, is a behavior very speciﬁc to the case F (x,y) = d(x,y)2. For more general functionals, we suspect that the complexity of M should have an impact as it is classically the case in nonparametric statistics or learning theory. Note in particular that whenever parameters α = β = 1 in (A2) and (A3), the rate vn in Theorem 2.5 becomes

− 2 n 2+D if D< 2, vn =  (log n)/√n if D = 2, − 1  n D if D> 2, which corresponds to known state of the art learning rates, under complexity assumptions in the same ﬂavor as (B2), as displayed for instance by Theorem 2 in [RST17]. However, exact situations under which Theorem 2.5 provides optimal rates of convergence remains unclear to us.

Finally, note that the second inequality in both Theorems 2.1 and 2.5 hold for the limiting case β = 0 (with A remaining ﬁnite), which correspond to dropping assumption (A3). In the context of Theorem 2.1 (or that of Theorem 2.5 with D< 2α) the case β = 0 gives rise to a bound of order

C (F (x ,y) F (x∗,y)) dP (y) , n − ≤ √n ZM with high probability. Note however that this limiting case does not allow to provide any bound for ∗ d(xn,x ).

3 Variance inequalities

This section studies conditions implying the validity of (A3). The ﬁrst three paragraphs below focus on the barycenter problem, i.e. the case where F (x,y) = d(x,y)2, and investigate (A3) in the light of curvature bounds. Aleksandrov curvature bounds of a geodesic space (see paragraph A.3) is a key concept of comparison geometry and many geometric phenomena are known to depend on whether the space M has a curvature bounded from below or above. In paragraphs 3.1 and 3.2, devoted respectively to non-positively and non-negatively curved spaces, we show that curvature bounds also aﬀect statistical procedures through their relation with (A3). Finally, paragraph 3.3 addresses the case of a general F and connects (A3) to its convexity properties. The material presented in this section relies heavily on background in metric geometry gathered in appendix A for convenience.

3.1 Non positive curvature This ﬁrst paragraph introduces a fundamental insight due to K.T. Sturm, in the context of geodesic spaces of non-positive curvature, that has strongly inﬂuenced our study. To put the following result in perspective, we recall that a geodesic space (M, d) is said to have non-positive curvature (curv(M) 0 ≤ for short) if, for any p,x,y M and any geodesic γ : [0, 1] M such that γ(0) = x and γ(1) = y, ∈ → d(p, γ(t))2 (1 t)d(p,x)2 + td(p,y)2 t(1 t)d(x,y)2, ≤ − − − for all t [0, 1]. Non-positive curvature is given a probabilistic description in the next result. ∈

9 Theorem 3.1 (Theorem 4.9 in [Stu03]). Let (M, d) be a separable and complete metric space. Then, the following properties are equivalent.

(1) (M, d) is geodesic and curv(M) 0. ≤ (2) Any probability measure Q (M) has a unique barycenter x∗ M and, for all x M, ∈ P2 ∈ ∈ d(x,x∗)2 (d(x,y)2 d(x∗,y)2)dQ(y). (3.1) ≤ − ZM In words, Theorem 3.1 states in particular that (A3) holds for any possible probability measure P on M, with K = 1 and β = 1, provided curv(M) 0. It is worth mentioning again that (A1) and (A2) 3 ≤ also hold, provided in addition diam(M) < + , so that the case of bounded metric spaces with non- ∞ positive curvature ﬁts very well our basic assumptions. Condition curv(M) 0 is satisﬁed in a number of ≤ interesting examples. Such examples include (convex subsets of) Hilbert spaces or the case where (M, d) is a simply connected Riemannian manifold with non-positive sectional curvature. Other examples are metric trees and other metric constructions such as products or gluings of spaces of non-positive curvature (see [BH99], [BBI01] or [AKP17] for more details).

3.2 Non negative curvature and extendable geodesics The present paragraph investigates the case of spaces of non-negative curvature. Contrary to the case of spaces of non-positive curvature, condition (A3) may not hold for every probability measure P on M if curv(M) 0. Indeed, note that unlike in the case when curv(M) 0, there might exist probability ≥ ≤ measures P (M) with more than one barycenter whenever curv(M) 0. A simple example when ∈ P2 ≥ M = Sd−1, the unit euclidean sphere in Rd with angular metric, is the uniform measure on the equator having the north and south poles as barycenters. Since (A3) implies uniqueness of barycenter x∗, this condition disqualiﬁes such probability measures. Hence, establishing conditions under which (A3) holds is more delicate whenever curv(M) 0. The next result provides an important ﬁrst step in this direction. ≥ Theorem 3.2. Let (M, d) be a separable and complete geodesic space such that curv(M) 0. Let P ≥ ∈ (M) and x∗ be a barycenter of P . Then, for all x M, P2 ∈

∗ 2 x 2 ∗ 2 d(x,x ) k ∗ (y)dP (y)= (d(x,y) d(x ,y) )dP (y), (3.2) x − ZM ZM where, for all x = x∗ and all y, 6 2 2 x logx∗ (x) logx∗ (y) x∗ d(x,y) k ∗ (y) = 1 k − k − . (3.3) x − d(x,x∗)2

Therefore, P satisﬁes (A3) with K = 1/k and β = 1 if and only if, for all x M, 3 ∈

x k k ∗ (y)dP (y). (3.4) ≤ x ZM By deﬁnition of a barycenter, the right hand side of (3.2) is non negative. In addition curv(M) 0 ≥ implies that d(x,y) logp(x) logp(y) p for all x,y,p M (see Proposition A.13 in appendix A). ≤ k − k x ∈ Combining these two observations with the deﬁnition of kx∗ (y) implies that

x 0 k ∗ (y)dP (y) 1. ≤ x ≤ ZM The next result identiﬁes a condition under which a variance inequality holds.

10 Theorem 3.3. Let (M, d) be a separable and complete geodesic space such that curv(M) 0. Let P ≥ ∈ (M) and x∗ be a barycenter of P . Fix λ> 0 and suppose that the following properties hold. P2 (1) For P -almost all y M, there exists a geodesic γ : [0, 1] M connecting x∗ to y that can be ∈ y → extended to a function γ+ : [0, 1+ λ] M that remains a shortest path between its endpoints. y → ∗ (2) The point x remains a barycenter of the measure Pλ = (eλ)#P where eλ : M M is defined by + → eλ(y)= γy (1 + λ). Then, for all x M, ∈ 1+ λ d(x,x∗)2 (d(x,y)2 d(x∗,y)2)dP (y), ≤ λ − ZM and thus (A3) holds with K3 = (1+ λ)/λ and β = 1. Examples of geodesic spaces of non-negative curvature include (convex subsets of) Hilbert spaces or the case where (M, d) is a simply connected Riemannian manifold with non-negative sectional curvature. Next is a simple example where the condition (1) of Theorem 3.3, i.e. the ability to extend geodesics, takes a simple form. Example 3.4 (Unit sphere). Let M = Sd−1 be the unit Euclidean sphere in Rd equipped with the angle metric. Let P (M) be such that it has a unique barycenter x∗ M. In M, a shortest path between ∈ P2 ∈ two points is a part of a great circle and a part of a great circle is a shortest path between its endpoints if, and only if, it has length less than π. Therefore, if a neighborhood V of C(x∗)= x∗ , the cut locus {− } of x∗, satisfies P (V ) = 0, then condition (1) of Theorem 3.3 is satisfied for some λ > 0. Note however that condition (1) is not enough to give a variance inequality in general. Indeed, consider the uniform measure P on the equator with the north and south poles for barycenters. Then, for P -almost all y M, ∈ the geodesic connecting the south pole x∗ to y can be extended by a factor 1+ λ = 2 in the sense of (1) in the above theorem. However, since there is no unique barycenter, no variance inequality can hold in this case. Therefore, requirement (2) cannot be dropped in Theorem 3.3. In the rest of this paragraph, we provide a sufficient condition for the extendable geodesics condition (1) of Theorem 3.3 to hold in the context where (M, d) = ( (H),W ) is the Wasserstein space over a P2 2 Hilbert space H. In this case, (M, d) is known to have non-negative curvature (see Section 7.3 in [AGS08]). We recall the following definition. For a convex function φ : H R, its subdifferential ∂φ H2 is defined → ⊂ by (x,y) ∂φ z H, φ(z) φ(x)+ y, z x . ∈ ⇔∀ ∈ ≥ h − i Then we can prove the following result. Theorem 3.5. Let (M, d) = ( (H),W ) be the Wasserstein space over a Hilbert space H. Let µ and ν P2 2 be two elements of M and let γ : [0, 1] S be a geodesic connecting µ to ν in M. Then, γ can be extended → by a factor 1+ λ (in the sense of (1) in Theorem 3.3) if, and only if, the support of the optimal transport plan π of (µ,ν) lies in the subdifferential ∂φ of a λ -strongly convex map φ : H R. 1+λ → Again, like in example 3.4, this gives a condition that ensures the validity of (1) in Theorem 3.5, but it can be shown that it is not enough to obtain a variance inequality.

3.3 Convexity Here, we connect (A3) to convexity properties of functional F along paths in M. Deﬁnition 3.6 ((k, β)-convexity). Given k > 0, β (0, 1] and a path γ : [0, 1] M, a function ∈ → f : M R is called (k, β)-convex along γ if the function → 2 t [0, 1] f(γ(t)) kd(γ(0), γ(1)) β t2 ∈ 7→ −

11 is convex. If (M, d) is geodesic, a function f : M R is called (k, β)-geodesically convex if, for all → x,x′ M, f is (k, β)-convex along at least one geodesic connecting x to x′. ∈ In the sequel, we abbreviate (k, 1)-convexity by k-convexity. When M is geodesic, a (k, β)-convex function f : M R refers to a (k, β)-geodesically convex function unless stated otherwise. Note that → (k, β)-convexity is a special case of uniform convexity (see Deﬁnition 1.6. in [Stu03]). We start by a general result.

Theorem 3.7. Let k > 0 and β (0, 1]. Suppose that, for all x M, there exists a path connecting x∗ ∈ ∈ to x along which the function x M F (x,y)dP (y) ∈ 7→ ZM is (k, β)-convex. Then, for all x M, ∈ 1 β d(x,x∗)2 (F (x,y) F (x∗,y)) dP (y) , (3.5) ≤ k − ZM −β and hence (A3) holds for K3 = k . In particular, the assumption of the theorem holds whenever, for all x M and (P -almost) all y M, there exists a path connecting x∗ to x along which the function ∈ ∈ F (., y) is itself (k, β)-convex. A special case in when (M, d) is geodesic and, for (P -almost) all y M, the ∈ functional x M F (x,y) is itself (k, β)-geodesically convex. ∈ 7→ The previous result is deliberately stated in a general form. This allows to investigate several notions of convexity that may coexists as in the space of probability measures.

Remark 3.8. For some spaces, such as the Wasserstein space over Rd, there exist two canonical paths between two points µ and ν. One is the geodesic γ deﬁned by the fact that W (γ , γ )= s t W (µ,ν) for t 2 t s | − | 2 s,t [0, 1] (which needs not be unique). A second one is the linear interpolation between the two measures ∈ ℓ = (1 t)µ + tν. It may be that x F (x,y)dP (y) is strongly convex along only one of these two paths. t − 7→ This case is further discussed in section 4. R We end by discussing the notion of a k-convex metric space. In the remainder of the paragraph, we consider F (x,y)= d(x,y)2.

Deﬁnition 3.9. A geodesic space (M, d) is said to be k-convex if, for all y M, the function x M ∈ ∈ 7→ d(x,y)2 is k-geodesically convex.

Using Theorem 3.7, it follows that if M is k-convex, then (A3) holds with β = 1 and K3 = 1/k. When k = 1, note that a geodesic space is k-convex if and only if curv(M) 0 (see Proposition A.6). When ≤ k = 1, the connection between curvature bounds and k-convexity of a metric space is not straightforward. 6 In particular, k-convex metric spaces include many interesting spaces, for which condition curv(M) 0 ≤ does not hold. We give two examples.

Example 3.10 (Proposition 3.1. in [Oht07]). Let M be a geodesic space with curv(M) 1 and diam(M) < ≤ π/2. Then M is k-convex for π k = 4 diam(M) tan diam(M) . 2 − Implication of this result can be compared to Theorem 3.3. In the context of M S2, considered in ⊂ example 3.4, the above result states that M is k-convex provided it is included in the interior of an 1/8th of a sphere. In comparison, Theorem 3.3 expresses that a variance inequality (A3) may hold for a measure supported on the whole sphere minus a neighbourhood of the cut locus of the barycenter.

Example 3.11 (Theorem 1 in [BCL95]). If (S, ,µ) is a measured space and p (1, 2], then M = Lp(µ) S ∈ is (p 1)-convex. −

12 4 Further examples

We describe additional examples, diﬀerent from the usual barycenter problem, where our results apply. In these examples, we focus mainly on functionals over the Wasserstein space. Paragraphs 4.1 addresses the case of f-divergences. Subsection 4.2 discusses interaction energy. In 4.3 we consider several examples related to the approximation of barycenters.

4.1 f-divergences We call f-divergence a functional of the form

dµ f dν dν if µ ν, Df (µ,ν) := ≪ (4.1) ( + otherwise, R ∞ for some convex function f : R R. Such functionals are also known as internal energy, relative + → functionals or Csiszár divergences. Speciﬁc choices for function f give rise to well known examples. For instance, f(x)= x log x gives rise to the Kullback-Leibler divergence (or relative entropy). Minimisers of the average Kullback-Leibler divergence, or its symmetrised version, have been considered for instance in [Vel02] for speech synthesis. Other functions like f(x) = (x 1)2 or f(x) = x 1 /2 lead respectively − | − | to the chi-squared divergence or the total variation. The next results present suﬃcient conditions under which (A1), (A2) and (A3) hold in this case. First, suppose M (E) where E Rd is a convex set. ⊂ P2 ⊂ Note that (A1) holds if there exists 0 < c− < c < + and a reference measure such that all µ M have + ∞ ∈ a density gµ with respect to this measure with values in [c−, c+] on their support. Next, we show that (A2) holds under related conditions.

Theorem 4.1. Suppose that all measures µ M have a density g with respect to some reference measure. ∈ µ Suppose there exist 0 < c− < c < + such that all g take values in [c−, c ]. Suppose in addition that + ∞ µ + there exists Λ > 0 such that all gµ are Λ-Lipschitz on E. Assume ﬁnally that f is diﬀerentiable and that f ′ is L-Lipschitz on E. Then, for all ν,µ,µ′ M, ∈ ′ 2LΛc+ ′ Df (µ,ν) Df (µ ,ν) 2 W2(µ,µ ), | − |≤ c−

2 so that (A2) holds for K2 = 2LΛc+/c− and α = 1. The next result is devoted to condition (A3) and is a slight modiﬁcation of Theorem 9.4.12 in [AGS08]. Below we say that ν M is λ-log-concave if dν(x)= e−U(x)dx for some λ-convex U : E R. ∈ → Theorem 4.2. Consider h(s)= esf(e−s). Suppose that h is convex and that there exists c> 0 such that, for all u R and all ε > 0, h(u) h(u + ε) cε. Suppose ﬁnally that M is a geodesically convex subset ∈ − ≥ of (E). Then the following holds. P2 (1) For any λ-log-concave ν M, the functional µ M D (µ,ν) is cλ-geodesically convex. ∈ ∈ 7→ f (2) Let λ > 0 and suppose P (M) is supported on λ-log-concave measures in M. Then for any ∈ P minimizer µ∗ arg min D (µ,ν)dP (ν), ∈ f µ∈M ZM and any µ M, ∈ 1 W (µ,µ∗)2 (D (µ,ν) D (µ∗,ν)) dP (ν), 2 ≤ cλ f − f ZM and thus (A3) holds with K3 = 1/cλ and β = 1.

13 Note that the requirements made on f in Theorem 4.2 are compatible with f(x) = x log x for c = 1. However, these requirements exclude examples such as f(x) = (x 1)2. At the price of a smaller exponent − β = 1/2, the next result shows that (A3) holds for any choice of a strongly convex f. Theorem 4.3. Let f : R R be k-convex. Then the following holds. + → (1) Let ν,µ ,µ (E) be such that µ ν and µ ν. Suppose in addition that 0 1 ∈ P2 0 ≪ 1 ≪ m (ν) := x 4 dν(x) < + . 4 k k ∞ Z k 1 Then, Df (., ν), is ( , )-convex along the linear interpolation ℓt = (1 t)µ0 + tµ1. 4m4(ν) 2 − (2) Let m> 0. Suppose that M is a linearly convex subset of (E) and suppose P (M) is supported P2 ∈ P on measures ν M such that m (ν) m. Then, for any minimizer ∈ 4 ≤ µ∗ arg min D (µ,ν)dP (ν), ∈ f µ∈M ZM and any µ M, ∈ m 1/2 W (µ,µ∗)2 2 (D (µ,ν) D (µ∗,ν)) dP (ν) , 2 ≤ k f − f ZM 1/2 and thus (A3) holds with K3 = 2(m/k) and β = 1/2. We end this paragraph with an important remark. Using Theorem 3.7, and as in Theorem 4.2, condition (A3) can be deduced from k-convexity of the f-divergence. Strong convexity of the f-divergence was used by Sturm, Lott and Villani (see [Stu06a, LV09] and references therein) to deﬁne the celebrated synthetic notion of Ricci curvature bounds. In particular, for an underlying Riemannian D-dimensional manifold E equipped with a measure with density e−V with respect to the Riemannian measure, k-convexity of such f-divergence is equivalent to Ric + Hess(V ) k ≥ where Ric stands for the Ricci curvature tensor and Hess(V ) is the Hessian of V , given s sDf(s−D) is 7→ geodesically convex.

4.2 Interaction energy Interaction energy of a measure µ, for a given function g on M M called interaction potential, is usually × defined as the integral of g w.r.t. the product measure µ µ. Here, we keep the same terminology for a ⊗ larger class of functionals on the Wasserstein space. Let (E, ρ) be a separable, complete, locally compact and geodesic space. Let M = (E) be equipped with the Wasserstein metric W . Consider functional P2 2 I : M M R defined by g × → + Ig(µ,ν)= g(x,y)dµ(x)dν(y), ZE×E for a measurable g : E E R . Note first that (A1) is clearly satisfied provided g is upper bounded or × → + g is continuous and E is compact. The next two results present sufficient conditions for (A2) and (A3) to hold. Theorem 4.4. Fix L> 0 and suppose that g is L-Lipschitz in the first variable when the second is fixed. Then, for all µ,µ′,ν (E), ∈ P2 I (µ,ν) I (µ′,ν) LW (µ,µ′), | g − g |≤ 2 and (A2) holds with K2 = L and α = 1.

14 Theorem 4.5. Fix k 0 and β (0, 1]. Suppose that g is (k, β)-geodesically convex in the first variable ≥ ∈ when the second is fixed. Then, the following holds. (1) For any fixed ν (E), the functional I (., ν) is (k, β)-geodesically convex on (E). ∈ P2 g P2 (2) Let P ( (E)). Then for any minimizer ∈ P P2 µ∗ arg min I (µ,ν)dP (ν), ∈ g µ∈P2(E) Z and any µ (E), ∈ P2 1 β W (µ,µ∗)2 (I (µ,ν) I (µ∗,ν)) dP (ν) , 2 ≤ k g − g Z β and (A3) holds with K3 = 1/k .

4.3 Regularised Wasserstein distance In this final paragraph, we present a few functionals of interest whose study, in the light of the previous results, could provide interesting research perspectives. In certain cases, like for the Wasserstein space, the distance is difficult to compute. In the field of computational optimal transport, a lot of work has been devoted to construct computationally friendly approximations of the Wasserstein distance. Such approximations include the sliced Wasserstein distance, penalised Wasserstein distances or regularised Wasserstein distances (see [CP19] for more details). In some of these approximations, one hope is that one may enforce some form of convexity allowing for condition (A3) to be valid in a wide setting. The Sinkhorn divergence, defined below, is an approximation of the Wasserstein distance that has been widely used in practice due to its attractive computationally properties. Denote by D the relative entropy, i.e. D = D for f(x)= x log x where D is as in (4.1), and Γ(µ,ν) the set of probability measures π on Rd Rd f f × with marginals µ an ν respectively. Let µ,ν (Rd), and γ > 0. The entropy-regularised Wasserstein ∈ P2 distance or Sinkhorn divergence between µ and ν is defined by

2 2 Wγ(µ,ν) := inf d(x,y) dπ(x,y)+ γD(π,µ ν). (4.2) π∈Γ(µ,ν) ⊗ Z Identifying scenarios in which Wγ satisﬁes conditions (A1)-(A3) remains for us an open question. Note however that, using Lagrange multipliers, the Sinkhorn divergence can be alternatively expressed as

inf d(x,y)2 dπ(x,y) : π Γ(µ,ν), D(π,µ ν) δ , ∈ ⊗ ≤ Z for some δ 0. Letting δ = 0, the above expression reduces to the interaction energy Ig studied in para- ≥ 2 graph 4.2 for g(x,y)= d(x,y) . This observation suggests that, for large values of γ, functional Wγ(µ,ν) should satisfy (A1)-(A3) under reasonable conditions. An interesting alternative notion of regularised Wasserstein distance, via factored couplings, was introduced in [FHN+19]. In order to deﬁne this notion, we introduce some notation. For a partition = (C ,...,C ), and a measure µ, denote by µC the measure C 1 n i µ restricted to Ci. For n N, denote by Γn(µ,ν) the set of measures γ such that there exists two partition ∈ 0 1 0 and 1 such that γ = n λ µC νC , with λ := µ(C0) = ν(C1). For two measures µ and ν, the C C i=1 i i ⊗ i i i i regularised Wasserstein distance between µ and ν is deﬁned by P n n 2 2 2 RWn(µ,ν) := inf d dγ = inf λi d d(µi νi) : λiµi νi = γ Γn(µ,ν) . γ∈Γn(µ,ν) ( ⊗ ⊗ ∈ ) Z Xi=1 Z Xi=1 While the trivial case n = 1 reduces to the interaction energy studied above, it remains unclear to us in which setting our assumptions apply to this functional in general.

15 5 Proofs

5.1 Proof of Theorem 2.1 The proof is based on three auxiliary lemmas. The ﬁrst lemma provides an upper bound on the largest ﬁxed point of a random nonnegative function. The proof follows from a combination of arguments presented in Theorem 4.1, Corollary 4.1 and Theorem 4.3 in [Kol11].

Lemma 5.1. Let φ(δ) : δ 0 be non-negative random variables (indexed by all deterministic δ 0) { ≥ } ≥ such that, almost surely, φ(δ) φ(δ′) if δ δ′. Let b(δ, t) : δ 0,t 0 , be (deterministic) real numbers ≤ ≤ { ≥ ≥ } such that b(δ, t) b(δ, t′), as soon as t t′, and such that ≤ ≤ P(φ(δ) b(δ, t)) e−t. ≥ ≤ Finally, let δˆ be a nonnegative random variable, a priori upper bounded by a constant δ¯ > 0, and such that, almost surely, δˆ φ(δˆ). ≤ Then deﬁning, for all t 0, ≥ b δ, tδ b(t) := inf α> 0 : sup α 1 , ( δ≥α δ ≤ ) we obtain, for all t 0, ≥ P(δˆ b(t)) 2e−t. ≥ ≤ The second lemma is due to [Bou02] and improves upon the work of [Tal96] by providing explicit constants. Given a family of functions f : M R, denote F → 2 2 P Pn F := sup(P Pn)f and σF := sup P (f Pf) , | − | f∈F − f∈F −

1 n where M and P are as deﬁned in paragraph 2.1 and Pn = n i=1 δYi . Lemma 5.2. Suppose that all functions in are [a, b]-valued,P for some a < b. Then, for all n 1 and F ≥ all t> 0, E 2t 2 E (b a)t P Pn F P Pn F + σF + 2(b a) P Pn F + − , | − | ≤ | − | r n − | − | 3n with probability larger than 1 e−t. − For background on empirical processes, including the proof of Lemma 5.2, we refer the reader to [GN15]. Finally, the third result we need is the following generalized version of Dudley’s entropy bound (see, for instance, Theorem 5.31 in [vH16]).

Lemma 5.3. Let (Xt)t∈E be a real valued process indexed by a pseudo metric space (E, d). Suppose that the three following conditions hold. (1) (Separability) There exists a countable subset E′ E such that, for any t E, ⊂ ∈

Xt = lim Xt′ , a.s. t′→t,t′∈E′

(2) (Subgaussianity) For all s,t E, X X is subgaussian in the sense that ∈ s − t θ2d(s,t)2 θ R, log Eeθ(Xs−Xt) . ∀ ∈ ≤ 2

16 (3) (Lipschitz property) There exists a random variable L such that, for all s,t E, ∈ X X Ld(s,t), a.s. | s − t|≤ Then, for any S E and any ε 0, we have ⊂ ≥ +∞ E sup Xt 2εE[L] + 12 log N(S, d, u) du. t∈S ≤ ε Z p We are now in position to prove Theorem 2.1.

Proof of Theorem 2.1. (1) For any δ 0, denote ≥ M(δ) := x M : P (F (x, .) F (x∗, .)) δ , { ∈ − ≤ } and φ (δ) := sup (P P )(F (x, .) F (x∗, .)) : x M(δ) . n { − n − ∈ } As a consequence of Assumption (B1), the set M is separable. Hence, the quantity φn(δ) is measurable, as well as all suprema involved in the rest of the proof. Deﬁne

δ := P (F (x , .) F (x∗, .)). n n − By deﬁnition of x , P (F (x , .) F (x∗, .)) 0 so that n n n − ≤ δ (P P )(F (x , .) F (x∗, .)) φ (δ ). n ≤ − n n − ≤ n n As a result, in order to upper bound δn with high probability, it is enough to upper bound φn(δ), for ﬁxed δ 0, and apply Lemma 5.1. Denoting ≥ σ2(δ) := sup P (F (x, .) F (x∗, .))2 : x M(δ) , { − ∈ } and observing that 2K F (x,y) F (x∗,y) 2K , for all x,y M due to (A1), it follows from Lemma − 1 ≤ − ≤ 1 ∈ 5.2 that inequality

2t 2 4K1t φn(δ) Eφn(δ)+ (σ (δ) + 8K1 Eφn(δ)) + , ≤ r n 3n holds with probability at least 1 e−t. Using basic inequalities √u + v √u + √v and 2√uv u + v for − ≤ ≤ positive numbers, we further deduce that

2t 16K1t φn(δ) 2Eφn(δ)+ σ(δ) + , (5.1) ≤ r n 3n with probability at least 1 e−t. Combining (A2) and (A3), we deduce that for all x M, − ∈ P (F (x, .) F (x∗, .))2 K2Kα(P (F (x, .) F (x∗, .)))αβ , (5.2) − ≤ 2 3 − and therefore, σ2(δ) K2Kαδαβ. (5.3) ≤ 2 3 Next, we provide an upper bound for Eφn(δ). Let σ1,...,σn be a sequence of i.i.d. random signs, i.e. such that P(σ = 1) = P(σ = 1) = 1/2, independent from the Y ’s. The symmetrization principle (see, i − i i e.g., Lemma 7.4 in [vH16]) indicates that

n 1 Eφ (δ) 2E sup σ (F (x,Y ) F (x∗,Y )) : x M(δ) n ≤ n i i − i ∈ ( i=1 ) Xn 1 = 2E sup σiF (x,Yi) : x M(δ) , (n ∈ ) Xi=1

17 where the last line follows from the fact that the σi’s are centered and independent of the Yi’s. Now, introduce the set = F (x, .) : x M , and observe that, conditionally on the Y ’s, the process F { ∈ } i n 1 X := σ f(Y ), f , f √n i i ∈ F Xi=1 satisﬁes the separability condition of Lemma 5.3 due to the separability of M. In addition, this process is subgaussian since

2 2 θ dn(f, g) f, g , θ R, log E[eθ(Xf −Xg) Y ,...,Y ] , ∀ ∈ F ∀ ∈ | 1 n ≤ 2 where n 1 d (f, g)2 = (f(Y ) g(Y ))2 n n i − i Xi=1 2 is the natural metric in L (Pn). Also, it satisﬁes the Lipchitz condition X X √nd (f, g). | f − g|≤ n Hence, denoting (δ)= F (x, .) : x M(δ) and applying Lemma 5.3, we obtain F { ∈ } 12 +∞ Eφn(δ) 2E inf 2ε + log N( (δ), dn, u) du . ≤ ε≥0 √n F ε Z p But combining (A2) and (A3), we deduce that, almost surely,

1 u α N( (δ), d , u) N M(δ), d, F n ≤ K 2 ! 1 α ∗ β u N B(x , K3δ ), d, . ≤ K2 ! p As a result,

1 +∞ α 12 ∗ β u Eφn(δ) 2 inf 2ε + log N B(x , K3δ ), d, du ≥ v ≤ ε 0  √n ε u K2 !   Z u p  t K Kα/2δαβ/2 1  2 3 α 12 ∗ β u = 2inf 2ε + log N B(x , K3δ ), d, du ≥ v ε 0  √n ε u K2 !   Z u p  t K Kα/2δαβ/2 α/2  12 2 3 D CαK K δαβ/2  2 inf 2ε + log 2 3 du , ε≥0 v ≤  √n ε u α u !   Z u  t where the last inequality follows from (B1). Assuming without loss of generality thatC 1 and using the ≥ simple upper bound log(x) x 1 x, for all x> 0, it follows from straightforward computations that ≤ − ≤ α 1 α 2 48C 2 D 2 K2K αβ 1 E 3 2 − 2 φn(δ) 1 δ n . (5.4) ≤ α 2 · Combining (5.1),(5.3) and (5.4) implies therefore that we have

αβ D αβ t c3t φn(δ) bn(δ, t) := c1δ 2 + c2δ 2 + , ≤ r n rn n 18 with probability at least 1 e−t where − α α 96C 2 K K 2 α 16K 2 3 √ 2 1 c1 = 1 , c2 = 2K2K3 and c3 = . (5.5) α 2 3 Using the fact that δ φ (δ ) and Lemma 5.1, it follows that n ≤ n n

−1 tδ δn bn(t) := inf τ > 0 : sup δ bn δ, τ 1 , ≤ ( δ≥τ ≤ ) with probability larger that 1 2e−t. It therefore remains to provide an upper bound for b (t). Observing − n that αβ 1, and that for nonincreasing functions h : [0, + ) [0, + ) we have ≤ j ∞ → ∞

inf τ > 0 : h1(τ)+ + hm(τ) 1 max inf τ > 0 : hj(τ) 1/m , { · · · ≤ }≤ 1≤j≤m { ≤ } it follows that 1 1 2 D 2−αβ 2 t 2−αβ 3c3t b (t) max (3c ) 2−αβ , (3c ) 2−αβ , , n ≤ 1 n 2 n n ( ) where c , c , c are as in (5.5). To sum up, we have shown that, for all n 1 and all t> 0, 1 2 3 ≥ ∗ (F (xn,y) F (x ,y)) dP (y) M − Z 1 1 2 D 2−αβ 2 t 2−αβ 3c3t max (3c ) 2−αβ , (3c ) 2−αβ , , ≤ 1 n 2 n n ( ) with probability at least 1 2e−t where c , c , c are as in (5.5). Finally note that, at the price of a slightly − 1 2 3 worst dependence on the constants, the last term in the maximum can be removed. Indeed, if t

5.2 Proof of Example 2.3 We prove inequalities (2.5). The proof uses a classical strategy but is reproduced for completeness. Define αD : M (0, + ) [0, + ) by × ∞ → ∞ µ(B(x, r)) α (x, r)= , (5.6) D rD so that α− α (x, r) α by assumption. Let x M and 0 < ε r be fixed. Let x ,...,x be a ≤ D ≤ + ∈ ≤ 1 N minimal ε-net for B(x, r) of size N = N(B(x, r), d, ε). Then, by monotonicity and subadditivity of µ, it follows that N N α (x, r)rD = µ(B(x, r)) µ(B(x , ε)) = εD α (x , ε). D ≤ i D i Xi=1 Xi=1 D D By definition of α− and α , we deduce that α−r α Nε which proves the first inequality. To prove + ≤ + the second inequality, define the ε-packing number Npack(B(x, r), d, ε) of B(x, r) as the maximal number

19 m of points x ,...,x B(x, r) such that d(x ,x ) > ε for all i = j. A collection of such points is called 1 m ∈ i j 6 an ε-packing of B(x, r). It is a classical fact that the covering and packing numbers satisfy the duality property N(B(x, r), d, ε) N (B(x, r), d, ε) N(B(x, r), d, ε/2). ≤ pack ≤ In particular, to upper bound the ε-covering number of B(x, r) it suﬃces to upper bound its ε-packing number. Hence, let x1,...,xm be a maximal ε-packing of B(x, r) of size m = Npack(B(x, r), d, ε). Notice that the balls B(xi, ε/2), i = 1,...,m are disjoint by deﬁnition and included in B(x, r + ε/2). Hence, by monotonicity and additivity of µ, it follows that

m ε ε D ε ε αD x, r + r + = µ B x, r + µ B xi, 2 2 2 ≥ 2 ! i=1 m [ ε = µ B x , i 2 i=1 X m ε D ε = α x , . 2 D i 2 Xi=1 The deﬁnition of α− and α+ implies once again that ε D ε D mα− α r + , 2 ≤ + 2 and therefore α 2r D α 3r D m + + 1 + , ≤ α ε ≤ α ε − − which concludes the proof.

5.3 Proof of Theorem 2.5 The proof is identical to that of Theorem 2.1 up to a minor modiﬁcation. Here, the control on the complexity of set M is only global. Hence, the inequality

α/2 αβ/2 1 K2K3 δ α 12 ∗ β u Eφn(δ) 2 inf 2ε + log N B(x , K3δ ), d, du , ≥ v ≤ ε 0  √n ε u K2 !   Z u p  t used in the proof of Theorem 2.1, while still valid, cannot be exploited as such. We simply replace it by the upper bound

K Kα/2δαβ/2 1 12 2 3 u α Eφn(δ) 2 inf 2ε + log N M, d, du . ≥ v ≤ ε 0  √n ε u K2 !   Z u  t From then on, the proof is similar to that of Theorem 2.1. 

5.4 Proof of Theorem 3.2 The proof of Theorem 3.2 follows by combining Lemmas 5.7 and 5.8 below.

Deﬁnition 5.4 (Exponential barycenter). For any P (M), a point x∗ M is said to be an exponential ∈ P2 ∈ barycenter of P if

log ∗ (x), log ∗ (y) ∗ dP (x)dP (y) = 0. h x x ix ZM ZM 20 For the definition of log and ., . we refer the reader to Appendix A. Exponential barycenters where x h ix introduced in [ÉM91]. The definition of an exponential barycenter mimics that of the Pettis integral of a Hilbert valued function and stands as an alternative way to define the analog of the mean value of an element of (M). Next is an important property of exponential barycenters. P2 Theorem 5.5 (Theorem 45 in [Yok12]). Suppose that curv(M) 0. Let x∗ be an exponential barycenter −1 ≥ of P (M). Then the linear hull of the support of P log ∗ is isometric to a Hilbert space. ∈ P2 ◦ x Remark 5.6. Note that Theorem 45 of [Yok12] requires the tangent cone to be separable. It is not clear to us whether the tangent cone of a separable geodesic space with curvature bounded below is always separable. However, separability is used only to approximate (a countable number) of integrals w.r.t a measure by the same integral w.r.t. a finitely supported measure. This can be derived from the law of large number and thus separability is not necessary.

Using Theorem 5.5 we prove next that the identity of Theorem 3.2 holds for exponential barycenters.

Lemma 5.7. Suppose that curv(M) 0. Let P (M) and x∗ be an exponential barycenter of P . ≥ ∈ P2 Then, for all x M, ∈ ∗ 2 x 2 ∗ 2 d(x,x ) k ∗ (y)dP (y)= (d(x,y) d(x ,y) )dP (y), x − ZM ZM ∗ x where, for all x = x and all y, k ∗ (y) is as in (3.3). 6 x ∗ Proof of Lemma 5.7. Fix an exponential barycenter x of P . For brevity, denote log = logx∗ , . = . x∗ ∗ k k k k and ., . = ., . x∗ . Given any x M, let xt be a geodesic connecting x to x in M. It is an easy exercise h i h i ∈ ∗ to check that t log(x ) is a geodesic connecting log(x ) to log(x) in T ∗ M. According to Theorem 5.5, 7→ t x and the geometry of a Hilbert space, it follows that, for all x,y M and all t [0, 1], ∈ ∈ y x 2 = (1 t) y x∗ 2 + t y x 2 t(1 t) x x∗ 2, k − tk − k − k k − k − − k − k where here and throughout, we denote both a point x M and its image log(x) in the tangent cone T ∗ M ∈ x by the same symbol x when there is no risk of confusion. Now since curv(M) 0, we have d(x,y) x y ≥ ≤ k − k for all x,y M, with equality if x = x∗ or y = x∗. Hence, it follows from the previous identity that, for ∈ all x,y M and all t (0, 1), ∈ ∈ t(1 t)d(x,x∗)2 = (1 t)d(x∗,y)2 + t y x 2 y x 2 − − k − k − k − tk = t(d(x,y)2 d(x∗,y)2) + ( y x∗ 2 y x 2) − k − k − k − tk x ∗ 2 + t (1 k ∗ (y)) d(x,x ) . − x Reordering terms and dividing by t, we obtain

x ∗ 2 2 ∗ 2 1 ∗ 2 2 (k ∗ (y) t)d(x,x ) = (d(x,y) d(x ,y) )+ ( y x y x ). (5.7) x − − t k − k − k − tk Integrating with respect to P (dy), we obtain

∗ 2 2 x ∗ 2 2 ∗ 2 P ( x . xt . ) (P k ∗ (.) t)d(x,x ) = P (d(x, .) d(x , .) )+ k − k − k − k . x − − t Finally, using Theorem 5.5, we get that

P ( x∗ . 2 x . 2)= t2 x∗ x 2. k − k − k t − k − k − k Hence, letting t 0 in the previous identity leads to the desired result. →

21 Lemma 5.8. Suppose that curv(M) 0 and let P (M). Then a barycenter of P is an exponential ≥ ∈ P2 barycenter of P .

∗ Proof of Lemma 5.8. Fix a barycenter x of P . As in the proof of the previous Lemma, denote log = logx∗ , . = . ∗ and ., . = ., . ∗ for brevity. Also, we denote both a point x M and its image log(x) in k k k kx h i h ix ∈ the tangent cone T ∗ M by the same symbol x when there is no risk of confusion. Since curv(M) 0, the x ≥ ﬁrst statement of Theorem 45 in [Yok12] implies that

x,y dP (x)dP (y) 0. h i ≥ ZM ZM Moreover, we know that d(x,y) x y for all x,y M with equality if x = x∗ or y = x∗. In particular, ≤ k − k ∈ x∗ also minimises y x y 2 dP (x). 7→ k − k ZM Thus, for all y M, letting y be a geodesic connecting x∗ to y we get for all t (0, 1], ∈ t ∈ x 2 dP (x)= x x∗ 2 dP (x) k k k − k ZM ZM x y 2 dP (x) ≤ k − tk ZM = x t y 2 dP (x) k − · k ZM = ( x 2 2t x,y + t2 y 2) dP (x), k k − h i k k ZM where we have used the properties log and . stated in section A.5. Simplifying the above expression, we k k obtain, for all t (0, 1], ∈ 2 x,y dP (x) t y 2. h i ≤ k k ZM Letting t 0, we get → x,y dP (x) 0. h i ≤ ZM Integrating with respect to y, we obtain

x,y dP (x)dP (y) 0. h i ≤ ZM ZM Combining this observation with the ﬁrst inequality of the proof shows that x∗ is an exponential barycenter.

5.5 Proof of Theorem 3.3

+ Consider y in the support of P , denote yλ = γy (1 + λ) = eλ(y) and consider the map σy : [0, 1] M + ∗ → deﬁned by σy(t) = γy (t(1 + λ)). By assumption, σy is a geodesic connecting x to yλ. In addition, we have by construction that 1 σ (τ)= y where τ = . y 1+ λ

It follows from the properties of the map logy listed in appendix A that

y = (1 τ)x∗ + τy , − λ

22 where we identify a point u and its image log (u) in T M. Now since curv(M) 0 (see Proposition A.8), y y ≥ we know that curv(T M) 0 so that, for all x T M, y ≥ ∈ y x y 2 (1 τ) x x∗ 2 + τ x y 2 τ(1 τ) y x∗ 2 k − ky ≥ − k − ky k − λky − − k λ − ky λ 1 λ = x x∗ 2 + x y 2 y x∗ 2. 1+ λk − ky 1+ λk − λky − (1 + λ)2 k λ − ky

Using the fact that y x∗ 2 = (1+ λ)2 y x∗ 2 and the fact that d(u, v) u v , with equality if k λ − ky k − ky ≤ k − ky u = y or v = y, we deduce from the inequality above that λ λ d(x,x∗)2 x x∗ 2 1+ λ ≤ 1+ λk − ky 1 d(x,y)2 x y 2 + λd(x∗,y)2 ≤ − 1+ λk − λky 1 = (d(x,y)2 d(x∗,y)2) x y 2 +(1+ λ)d(x∗,y)2. − − 1+ λk − λky Integrating this inequality with respect to dP (y), it follows that

λ d(x,x∗)2 (d(x,y)2 d(x∗,y)2)dP (y) 1+ λ ≤ − ZM 1 + (1 + λ)d(x∗,y)2 x y 2 dP (y). − 1+ λk − λky ZM To conclude the proof, it remains to show that, for all x M, ∈ 1 ρ(x) := (1 + λ)d(x∗,y)2 x y 2 dP (y) 0. − 1+ λk − λky ≤ ZM Observing that (1 + λ)2d(x∗,y)2 = d(x∗,y )2 and that d(x,y ) x y , λ λ ≤ k − λky we deduce that 1 ρ(x) (d(x∗,y )2 d(x,y )2)dP (y) ≤ 1+ λ λ − λ ZM 1 = (d(x∗,y)2 d(x,y)2)dP (y), 1+ λ − λ ZM where we have used the fact that y = e (y) and the fact that P = (e )∗P . Hence, for all x M, λ λ λ λ ∈ inequality ρ(x) 0 follows from the fact that x∗ is a barycenter of P by assumption. ≤ λ 5.6 Proof of Theorem 3.5 We start by a technical lemma. Below H is a Hilbert space with scalar product ., . and associated norm h i . . k k Lemma 5.9. Let φ : H R be a convex function and ∂φ H2 its subdiﬀerential deﬁned by → ⊂ (x,y) ∂φ z H, φ(z) φ(x)+ y, z x . ∈ ⇔∀ ∈ ≥ h − i Then, for all c> 0,

(x,y) ∂φ z H, φ(z) φ(x)+ y, z x c x z 2. ∈ ⇔∀ ∈ ≥ h − i− k − k

23 Proof of the Lemma. One implication is obvious. For the second implication, suppose (x,y) H2 is such ∈ that z H, φ(z) φ(x)+ y, z x c z x 2. (5.8) ∀ ∈ ≥ h − i− k − k Then, on the one hand, we get by convexity of ϕ that, for all z H and all t (0, 1), ∈ ∈ 1 φ(z) φ(x) (φ(tz + (1 t)x) φ(x)). (5.9) − ≥ t − − On the other hand, applying (5.8), we obtain for all z H and all t (0, 1), ∈ ∈ φ(tz + (1 t)x) φ(x) t y, z x ct2 z x 2. (5.10) − − ≥ h − i− k − k Hence, combining (5.9) and (5.10), we deduce that for all z H and all t (0, 1), ∈ ∈ φ(z) φ(x) y, z x ct z x 2. − ≥ h − i− k − k Letting t 0 proves the other implication. → We are now in position to prove Theorem 3.5. Fix µ,ν S = (H) and denote γ : [0, 1] S a ∈ P2 → constant speed shortest path between µ and ν. By the Knott-Smith optimality criterion (see Theorem 2.12 in [Vil03]), π is an optimal transport plan of (µ,ν) if and only if its support lies in the graph of the subdiﬀerential of a convex function φ, i.e.

(x,y) suppπ z H, φ(z) φ(x)+ y, z x . (5.11) ∈ ⇒∀ ∈ ≥ h − i Suppose first that, for some λ > 0, γ : [0, 1] S can be extended to a function γ+ : [0, 1+ λ] S → → that remains a shortest path between its endpoints µ = γ+(0) = γ(0) and νλ := γ+(1 + λ). Then, by Theorem 7.2.2 in [AGS08], there exists an optimal transport plan πλ of (µ,νλ) such that π = Γ (µ,ν) { } o is that the law of λ 1 (X,Y ) := X, X + Y λ 1+ λ 1+ λ where (X,Y λ) πλ. In particular Y λ = (1+ λY ) λX. Therefore, there exists a convex function φλ ∼ − such that denoting yλ = (1+ λ)y λx − (x,y) suppπ (x,yλ) suppπλ ∈ ⇔ ∈ z H, φλ(z) φλ(x)+ yλ, z x ⇒∀ ∈ ≥ h − i z H, φλ(z) φλ(x)+(1+ λ) y, z x λ x, z x ⇔∀ ∈ ≥ h − i− h − i λ λ 1 z H, φλ(z)+ z 2 φλ(x)+ x 2 +(1+ λ) y, z x + λ x z 2 ⇔∀ ∈ 2 k k ≥ 2 k k h − i 2k − k φλ(z) λ z 2 φλ(x) λ x 2 z H, + k k + k k + y, z x ⇒∀ ∈ 1+ λ 2(1 + λ) ≥ 1+ λ 2(1 + λ) h − i z H, φ(z) φ(x)+ y, z x , (5.12) ⇔∀ ∈ ≥ h − i where we denote φλ 1 λ φ = + . 2. 1+ λ 2 1+ λk k λ λ Thus, suppπ lies in the subdifferential of φ that is 1+λ -convex, since φ is convex. Conversely, suppose that there exists a λ -convex function φ such that suppπ for π Γ (µ,ν) lies in 1+λ ∈ o the subdifferential of φ. Denote (X,Y ) π and set Y λ = Y + λ(Y X) νλ and (X,Y λ) πλ. Then ∼ − ∼ ∼

24 πλ is an optimal transport plan between µ and νλ if and only if there exists a convex function φλ such that the suppπλ lies in ∂φλ. In that case,

(1 + λ)2 W 2(µ,νλ)= E Y λ X 2 = (1+ λ)2W 2(µ,ν)= W 2(ν,νλ), 2 k − k 2 λ2 2 so that by Lemma 7.2.1 of [AGS08], ν is in the shortest path joining µ to νλ, which is the desired result. It thus just remains to prove that there exists a convex function φλ such that suppπλ lies in ∂φλ. Set 1 φλ = (1+ λ)φ λ . 2. − 2 k k φλ is convex since φ is λ -convex. Then, denoting again yλ = (1+ λ)y λx, 1+λ − (x,yλ) suppπλ (x,y) suppπ ∈ ⇔ ∈ z H, φ(z) φ(x)+ y, z x ⇒∀ ∈ ≥ h − i φλ(z) λ z 2 φλ(x) λ x 2 z H, + k k + k k + y, z x ⇔∀ ∈ 1+ λ 2(1 + λ) ≥ 1+ λ 2(1 + λ) h − i 1 z H, φλ(z) φλ(x)+ yλ, z x λ x z 2. (5.13) ⇔∀ ∈ ≥ h − i− 2k − k Since φλ is convex, by Lemma 5.9, (5.13) is equivalent to

z H, φλ(z) φλ(x)+ yλ, z x , ∀ ∈ ≥ h − i that is, suppπλ lies in the subdiﬀerential of the convex function φλ.

5.7 Proof of Theorem 3.7 For all x M, denote ∈ V (x)= F (x,y)dP (y). ZM Now ﬁx x M and let γ : [0, 1] M be a path connecting x∗ to x and along which V is (k, β)-convex. ∈ → Then, for all t [0, 1], ∈ 2 V (γ ) (1 t)V (x∗)+ tV (x) kt(1 t)d(x,x∗) β . t ≤ − − − Reordering these terms we obtain, for all t (0, 1), ∈ ∗ ∗ 2 V (x) V (x ) V (x ) V (γt) kd(x,x∗) β − + − ≤ 1 t t(1 t) − − V (x) V (x∗) − , ≤ 1 t − since V (x∗) V (γ ) by deﬁnition of x∗. Letting t tend to 0 concludes the proof. ≤ t 5.8 Proof of Theorem 4.1 For any µ M (E), let g be the density of µ with respect to the reference measure m. Fix ∈ ⊂ P2 µ ν,µ,µ′ M and denote for brevity ∈ gµ gµ′ f(aµ(x)) f(aµ′ (x)) aµ(x)= (x), aµ′ (x)= (x) and δµ,µ′ (x)= − . g g a (x) a ′ (x) ν ν µ − µ

25 Then, we obtain

′ D (µ,ν) D (µ ,ν)= (f(a (x)) f(a ′ (x)))g (x)dm(x) f − f µ − µ ν Z f(a (x)) f(a ′ (x)) = µ − µ (dµ(x) dµ′(x)) a (x) a ′ (x) − Z µ − µ ′ = δ ′ (x)(dµ(x) dµ (x)). (5.14) µ,µ − Z ′ Let us prove that δ ′ is Lipschitz. To that aim, observe that since f is L-Lipschitz, for all x,y E, µ,µ ∈ 1 ′ ′ δ ′ (x) δ ′ (y) = f ((1 t)a (x)+ ta ′ (x)) f ((1 t)a (y)+ ta ′ (y)) dt | µ,µ − µ,µ | | − µ µ − − µ µ | Z0 1 ′ ′ f ((1 t)aµ(x)+ taµ′ (x)) f ((1 t)aµ(y)+ taµ′ (y)) dt ≤ 0 | − − − | Z 1 L (1 t)(a (x) a (y)) + t(a ′ (x) a ′ (y)) dt ≤ | − µ − µ µ − µ | Z0 L max a (x) a (y) , a ′ (x) a ′ (y) . (5.15) ≤ {| µ − µ | | µ − µ |} Then, we see that

gµ(x)gν (y) gµ(y)gν (x) aµ(x) aµ(y)= − − gν(x)gν (y) g (x)(g (y) g (x)) + g (x)(g (x) g (y)) = µ ν − ν ν µ − µ , gν(x)gν (y) which implies, under the conditions of the theorem, that

2Λc+ aµ(x) aµ(y) 2 x y . (5.16) | − |≤ c− k − k

2 Combining (5.15) and (5.16) we deduce that δµ,µ′ is Lipschitz with constant at most 2LΛc+/c−. Using (5.14) and the Kantorovich-Rubinstein formula (see remark 6.5 in [Vil08]) we therefore obtain that

′ 2LΛc+ ′ Df (µ,ν) Df (µ ,ν) 2 W1(µ,µ ). | − |≤ c−

The result follows by observing that W (µ,µ′) W (µ,µ′) (see remark 6.6 in [Vil08]). 1 ≤ 2 5.9 Proof of Theorem 4.2 First note that the second statement of the theorem follows by Theorem 3.7. Hence, we need only to prove that, for all µ ,µ M, there exists a geodesic µ connecting µ to µ such that, for all 0 t 1, 0 1 ∈ t 0 1 ≤ ≤ D (µ ,ν) (1 t)D (µ ,ν)+ tD (µ ,ν) cλt(1 t)W (µ ,µ )2. f t ≤ − f 0 f 1 − − 2 0 1 If either µ or µ is not absolutely continuous with respect to ν, then the right hand side is + and 0 1 ∞ the inequality trivially holds. Suppose now that µ ν and µ ν. Since ν, and therefore µ , has a 0 ≪ 1 ≪ 0 density with respect to the Lebesgue measure, there exists an optimal transport map T : E E pushing → µ to µ . Letting T (x) = (1 t)x + tT (x), the curve µ = (T ) µ deﬁnes a geodesic connecting µ to µ 0 1 t − t t # 0 0 1 in M. For all t [0, 1], we denote ∈ dµ ρ (x)= t (x), t dx

26 so that, letting dν(x)= e−U(x)dx, U(x) dµt(x)= ρt(x)e dν(x). Letting DT (x) = (1 t)I + tDT (x) denote the diﬀerential of T at x, the change of variables formula t − t

ρ0(x)= ρt(Tt(x))det(DTt(x)), implies that

U(x) −U(x) Df (µt,ν)= f(ρt(x)e )e dx Z U(Tt(x)) ρ0(x)e det(DTt(x)) = f ρ0(x)dx det(DT (x)) ρ (x)eU(Tt(x)) Z t ! 0

= h(s(t,x))ρ0(x)dx, Z where s(t,x)= U(T (x)) + log det(DT (x)) log ρ (x). (5.17) − t t − 0 The transport map T being the gradient of a convex function, the map t [0, 1] log det(DT (x)) is ∈ 7→ t concave for any ﬁxed x E. The λ-convexity of U therefore implies that, for all t [0, 1], ∈ ∈ s(t,x) (1 t)s(0,x)+ ts(1,x)+ λt(1 t) x T(x) 2. ≥ − − k − k Using the assumptions on h, which imply in particular that it is decreasing, we deduce that

h(s(t,x)) h((1 t)s(0,x)+ ts(1,x)+ λt(1 t) x T(x) 2) ≤ − − k − k h((1 t)s(0,x)+ ts(1,x)) cλt(1 t) x T(x) 2 ≤ − − − k − k (1 t)h(s(0,x)) + th(s(1,x)) cλt(1 t) x T(x) 2. ≤ − − − k − k

Integrating the last inequality with respect to ρ0(x)dx yields

D (µ ,ν) (1 t)D (µ ,ν)+ tD (µ ,ν) cλt(1 t)W 2(µ ,µ ), f t ≤ − f 0 f 1 − − 2 0 1 which is the desired result.

5.10 Proof of Theorem 4.3 Denote dµ = g dν. The linear interpolation ℓ = (1 t)µ + tµ therefore satisﬁes i i t − 0 1 dℓ t = (1 t)g + tg . dν − 0 1 It follows from the k-convexity of f that

dℓ D (ℓ ,ν)= f t dν f t dν Z = f((1 t)g + tg )dν − 0 1 Z (1 t) f(g )dν + t f(g )dν kt(1 t) g g 2 dν ≤ − 0 1 − − | 1 − 0| Z Z Z = (1 t)D (µ ,ν)+ tD (µ ,ν) kt(1 t) g g 2 dν. − f 0 f 1 − − | 1 − 0| Z

27 According to Theorem 6.15 in [Vil08], we know that

W (µ ,µ )2 2 x 2 g (x) g (x) dν(x) 2 0 1 ≤ k k | 1 − 0 | Z 1/2 2m (ν)1/2 g (x) g (x) 2 dν(x) . ≤ 4 | 1 − 0 | Z Hence, we deduce that

k 4 Df (ℓt,ν) (1 t)Df (µ0,ν)+ tDf (µ1,ν) t(1 t)W2(µ0,µ1) , ≤ − − 4m4(ν) − which proves the ﬁrst statement. The second statement follows directly from Theorem 3.7.

5.11 Proof of Theorem 4.4 For ﬁxed y E, the fact that x E g(x,y) is L-Lipschitz, the Kantorovich-Rubinstein formula ∈ ∈ 7→ (see remark 6.5 in [Vil08]) and the fact that W W (see remark 6.6 in [Vil08]) implies that, for all 1 ≤ 2 µ,µ′ (E), ∈ P2 g(x,y)(dµ(x) dµ′(x)) LW (µ,µ′) LW (µ,µ′). | − |≤ 1 ≤ 2 Z Integrating with respect to ν (E) implies that ∈ P2 I (µ,ν) I (µ′,ν) g(x,y)(dµ(x) dµ′(x)) dν(y) | g − g |≤ | − | Z Z LW (µ,µ′). ≤ 2 5.12 Proof of Theorem 4.5 Given that (E, ρ) is separable, complete, locally compact and geodesic, the space ( (E),W ) is also P2 2 geodesic according to Corollary 7.22 in [Vil08]. In addition, given µ ,µ (E), a curve (µ ) ≤ ≤ in 0 1 ∈ P2 t 0 t 1 (E) is a geodesic connecting µ to µ if and only if there exists a probability measure Π on the set (E) P2 0 1 G of all geodesics γ : [0, 1] E in E such that → µt = (evalt)∗(Π), where eval (γ) = γ , for all γ (E), and where (eval , eval )∗(Π) is an optimal coupling of µ and µ . t t ∈ G 0 1 0 1 For such a geodesic,

Ig(µt,ν)= g(x,y)dµt(x)dν(y) ZZ = g(γt,y) dΠ(γ)dν(y) ZZ 2 (1 t)g(γ ,y)+ tg(γ ,y) kt(1 t)ρ(γ , γ ) β dΠ(γ)dν(y) ≤ − 0 1 − − 0 1 ZZ 2 = (1 t)I (µ ,ν)+ tI (µ ,ν) kt(1 t) ρ(γ , γ ) β dΠ(γ) − g 0 g 1 − − 0 1 Z 1 β (1 t)I (µ ,ν)+ tI (µ ,ν) kt(1 t) ρ(γ , γ )2 dΠ(γ) ≤ − g 0 g 1 − − 0 1 Z 2 = (1 t)I (µ ,ν)+ tI (µ ,ν) kt(1 t)W (µ ,µ ) β . − g 0 g 1 − − 2 0 1 This completes the proof of (1). Statement (2) follows directly by combining the ﬁrst statement and Theorem 3.7.

28 A Metric geometry

A.1 Geodesic spaces Let (M, d) be a metric space. We call path in M a continuous map γ : I M defined on an interval → I R. The length L(γ) [0, + ] of a path γ : I M is defined by ⊂ ∈ ∞ → n−1 L(γ) := sup d(γ(ti), γ(ti+1)), Xi=0 where the supremum is taken over all n 1 and all t t in I. A path is called rectifiable if it has ≥ 0 ≤···≤ n finite length. Two paths γ and γ are said to be equivalent if γ ϕ = γ ϕ for non-decreasing and 1 2 1 ◦ 1 2 ◦ 2 continuous functions ϕ1 and ϕ2. In this case, γ1 is said to be a reparametrisation of γ2 and we check that L(γ )= L(γ ). A path γ : [a, b] M is said to have constant speed if for all a s t b, 1 2 → ≤ ≤ ≤ t s L(γ )= − L(γ), (A.1) [s,t] b a − where γ[s,t] denotes the restriction of γ to [s,t]. Proposition A.1. Any rectifiable path has a constant speed reparametrisation γ : [0, 1] M. → Given x,y M, a path γ : [a, b] M is said to connect x to y if γ(a) = x and γ(b) = y. By ∈ → construction of the length function L, d(x,y) L(γ) for any path γ connecting x to y. The space M is ≤ called a length space if, for all x,y M, ∈ d(x,y) = inf L(γ), (A.2) γ where the infimum is taken over all paths γ connecting x to y. A length space is said to be a geodesic space if, for all x,y M, the infimum on the right hand side of (A.2) is attained. ∈ Definition A.2. In a geodesic space, we call geodesic between x and y any constant speed reparametrisation γ : [0, 1] M of a path attaining the infimum in (A.2). → For a geodesic γ, it follows from its minimising properties that

d(γ(s), γ(t)) = L(γ[s,t]), for all 0 s t 1. In particular, (A.1) translates in this case as ≤ ≤ ≤ d(γ(s), γ(t)) = (t s)d(γ(0), γ(1)), − for all 0 s t 1. We end by a general characterization of geodesic spaces. ≤ ≤ ≤ Proposition A.3. Let (M, d) be a metric space.

(1) If M is a geodesic space, then any two points x,y M admit a midpoint, i.e. a point z M such ∈ ∈ that 1 d(x, z)= d(y, z)= d(x,y). 2 (2) Conversely, if M is complete and if any two points in M admit a midpoint, then M is a geodesic space.

29 A.2 Model spaces Given a real number κ R, a geodesic space of special interest is the (complete and simply connected) ∈ 2-dimensional Riemannian manifold with constant sectional curvature κ. For given κ R, this metric 2 ∈ space (Mκ , dκ) is unique up to an isometry, and modelled as follows. If κ< 0, (M 2, d ) is the hyperbolic plane with metric multiplied by 1/√ κ. • κ κ − If κ = 0, (M 2, d ) is the Euclidean plane equipped with its Euclidean metric. • 0 0 If κ> 0, (M 2, d ) is the Euclidean sphere in R3 of radius 1/√κ with the angular metric. • κ κ 2 The diameter ̟κ of Mκ is + if κ 0, ̟κ := ∞ ≤ π/√κ if κ> 0. For κ R, there is a unique geodesic connecting x to y in (M 2, d ) provided d (x,y) < ̟ . By convention, ∈ κ κ κ κ we call triangle in M 2 any set of three distinct points p,x,y M 2, with perimeter κ { }⊂ κ peri p,x,y := d (p,x)+ d (p,y)+ d (x,y) < 2̟ . { } κ κ κ κ Side lengths of triangle p,x,y are the numbers d (p,x), d (p,y) and d (x,y). Given a, b, c > 0 satisfying { } κ κ κ the triangle inequality and such that a + b + c< 2̟κ, there exists a unique (up to an isometry) triangle p,x,y in M 2 such that d (p,x) = a, d (p,y) = b and d (x,y) = c. The angle ∢κ(x,y) at p in { } κ κ κ κ p p,x,y M 2 is deﬁned by { }⊂ κ a2 + b2 c2 − if κ = 0, ∢κ 2ab cos p(x,y) :=   cκ(c) cκ(a) cκ(b)  − · if κ = 0, κ s (a)s (b) 6 · κ κ  ′ where a = dκ(p,x), b = dκ(p,y), c = dκ(x,y) and cκ := sκ with sin(r√κ)/√κ if κ> 0, s (r) := (A.3) κ sinh(r√ κ)/√ κ if κ< 0. − − 2 We end by observing that the angle is constant along geodesics in the model space (Mκ , dκ). Proposition A.4. Let κ R and p,x,y (M 2, d ) be a triangle. If γ and γ are geodesics from p to ∈ { }⊂ κ κ x y x and from p to y respectively, then for all (s,t) (0, 1]2, ∈ ∢κ ∢κ p (γx(s), γy(t)) = p (x,y).

A.3 Curvature In this section, we describe the notion of curvature bounds of metric spaces. Curvature bounds in general 2 metric spaces are defined by comparison arguments involving the model surfaces (Mκ , dκ) discussed in the previous section. The fundamental device allowing for this comparison is that of a comparison triangle. Given a metric space (M, d), we define a triangle in M as any set of three points p,x,y M. For { } ⊂ κ R, a comparison triangle for p,x,y in M 2 is an isometric embedding of p,x,y in M 2, i.e. a set ∈ { } κ { } κ p ,x ,y M 2 such that { κ κ κ}⊂ κ dκ(pκ,xκ)= d(p,x), dκ(pκ,yκ)= d(p,y) and dκ(xκ,yκ)= d(x,y). Such a comparison triangle always exists (and is unique up to an isometry) provided peri p,x,y := d(p,x)+ d(p,y)+ d(x,y) < 2̟ . { } κ We are now in position to define curvature bounds for geodesic spaces.

30 Definition A.5. Let κ R and (M, d) be a geodesic space. ∈ (1) We say that curv(M) κ if for any triangle p,x,y M satisfying peri p,x,y < 2̟ , any ≥ { } ⊂ { } κ comparison triangle p ,x ,y M 2, any geodesic γ joining x to y in M and any geodesic γ { κ κ κ} ⊂ κ κ joining x to y in M 2 , we have for all t [0, 1], κ κ κ ∈ d(p, γ(t)) d (p , γ (t)). (A.4) ≥ κ κ κ (2) We say that curv(M) κ if the above definition holds with opposite inequality in (A.4). ≤ The previous definition has a natural geometric interpretation: if curv(M) κ (resp. curv(M) κ) a ≥ ≤ triangle p,x,y looks thicker (resp. thiner) than a corresponding comparison triangle p ,x ,y in M 2. { } { κ κ κ} κ In the context of κ = 0, the above definition may be given an alternative form of practical interest. Proposition A.6. Let (M, d) be a geodesic space. Then curv(M) 0 if, and only if, for any points ≥ p,x,y M and any geodesic γ joining x to y, we have ∈ t [0, 1], d(p, γ(t))2 (1 t)d(p,x)2 + td(p,y)2 t(1 t)d(x,y)2. ∀ ∈ ≥ − − − We have curv(M) 0 if, and only if, the same statement holds with opposite inequality. ≤ The proof follows immediately from Definition A.5 by exploiting the geometry of the Euclidean plane. Note indeed that, whenever p,x,y R2 and R2 is equipped with the Euclidean metric . . , the { } ⊂ k − k unique geodesic from x to y is γ(t) = (1 t)x + ty and, for all t [0, 1], − ∈ p γ(t) 2 = (1 t) p x 2 + t p y 2 t(1 t) x y 2. k − k − k − k k − k − − k − k For κ = 0, an equivalent formulation of Definition A.5, given only in terms of the ambient metric d, is given 6 in the next subsection using the notion of angle. A (complete) geodesic space (M, d) with curv(M) κ ≤ for some κ 0 is sometimes called a CAT(κ) space in reference to contributions of E. Cartan, A.D. ≥ Alexandrov and V.A. Toponogov. A CAT(0) space is also referred to as an NPC (non positively curved) space or an Hadamard space. Similarly, M is also called an PC (positively curved) space if curv(M) 0. ≥ If (M, d) is a Riemannian manifold (complete for instance) with sectional curvature lower (resp. upper) bounded by κ at every point, then curv(M) κ (resp κ) in the sense of Definition A.5. It is worth ≥ ≤ noting that the previous definitions are of global nature as they require comparison inequalities to be valid for all triangles (that admit a comparison triangle in the relevant model space). Some definitions of curvature require the previous comparison inequalities to hold only locally. The local validity of these comparison inequalities is known, under suitable conditions depending on the value of κ, to imply their global validity. Results in this direction are known as globalisation theorems.

A.4 Angles and space of directions Angles, as defined below, allow to provide alternative characterisations of curvature bounds. Let (M, d) be a metric space and let κ R. Given a triangle p,x,y in M with peri p,x,y < 2̟ , we define the ∈ { } { } κ comparison angle ∢κ(x,y) [0,π] at p by p ∈ d(p,x)2 + d(p,y)2 d(x,y)2 − if κ = 0, 2d(p,x)d(p,y) ∢κ  cos p(x,y) :=  cκ(d(x,y)) cκ(d(p,x)) cκ(d(p,y))  − · if κ = 0, κ s (d(p,x))s (d(p,y)) 6 · κ κ  where cκ and sκ are as in (A.3). In other words, given any comparison triangle pκ,xκ,yκ of p,x,y in 2 { } { } Mκ , ∢κ ∢κ p(x,y)= pκ (xκ,yκ). This allows to give an equivalent definition of curvature lower bounds that has the advantage of making sense on arbitrary metric spaces, not necessarily geodesic.

31 Definition A.7 (Quadruple comparison). Let (M, d) be a metric space. Let κ R. We say that ∈ curv(M) κ, if for any four disctinct points p,x,y,z M such that every three points have a perimeter ≥ ∈ less than ̟κ, ∢κ(x,y)+ ∢κ(y, z)+ ∢κ(z,x) 2π. p p p ≤ Proposition A.8. If (M, d) is a geodesic space, then curvature lower bounds as defined in Definition A.5 and Definition A.7 are equivalent. Moreover, even if (M, d) is not geodesic, and satisfies Definition A.7, then, equation (A.4) is satisfied for all t [0, 1] and γ(t) M such that ∈ ∈ d(γ(0), γ(t))/t = d(γ(t), γ(1))/(1 t)= d(γ(1), γ(0)). − The next result presents a characterization of curvature bounds in terms of the monotonicity of the comparison angle.

Proposition A.9 (Angle monotonicity). Let (M, d) be a geodesic space and let κ R. Then curv(M) κ ∈ ≥ (resp. curv(M) κ), in the sense of Deﬁnition A.5, if and only if, for any triangle p,x,y in M and ≤ { } any geodesics γx and γy from p to x and from p to y respectively, the function

(s,t) [0, 1]2 ∢κ(γ (s), γ (t)), ∈ 7→ p x y is non-increasing (resp. non-decreasing) in each variable when the other is ﬁxed.

In a geodesic space (M, d), if p M and γ and γ are two geodesics connecting p to x and y ∈ x y respectively, we define ∢κ ∢κ (γx, γy) := lim p(γx(s), γy(t)), s,t→0 when it exists. It follows from Proposition A.9 that this limit exists provided curv(M) κ or curv(M) κ. ≥ ≤ It may be shown furthermore that this limit is independent of κ. Hence, whenever M has upper or lower curvature bound, we denote ∢(γx, γy) the angle between these two geodesics. Given a third geodesic γ : [0, 1] S such that γ (0) = p and γ (1) = z, we have the triangular inequality z → z z ∢(γ , γ ) ∢(γ , γ )+ ∢(γ , γ ), (A.5) x y ≤ x z z y so that ∢ defines a pseudo metric on the set G(p) of all geodesics emanating from p. Defining the equivalence relation on G(p) by α β ∢(α, β) = 0, the angle ∢ induces a metric (still denoted ∢) ∼ ∼ ⇔ on the quotient set G(p)/ and we call space of directions the completion (Σ , ∢) of (G(p)/ , ∢). An ∼ p ∼ element of Σp is called a direction.

A.5 Tangent cones Metric spaces considered so far have a priori no diﬀerentiable structure. In this context, an analog of a tangent space is provided by the notion of a tangent cone. This section shortly reviews this notion. Below, M denotes a geodesic space with lower or upper bounded curvature in the sense of Deﬁnition A.5.

Definition A.10 (Tangent cone). Let p M. The tangent cone T M at p is the Euclidean cone over the ∈ p space of directions (Σp, ∢). In other words, TpM is the metric space: Whose underlying set consists in equivalent classes in Σ [0, + ) for the equivalence relation • p × ∞ ∼ defined by (α, s) (β,t) ((s = 0 and t = 0) or (s = t and α = β)). ∼ ⇔ A point in T M is either the tip of the cone o , i.e. the class Σ 0 , or a couple (α, s) p p p × { } ∈ Σ (0, + ) (identified to the class (α, s) ). p × ∞ { }

32 Whose metric d is defined (without ambiguity) by • p d ((α, s), (β,t)) := s2 + t2 2st cos ∢(α, β). p − p For u = (α, s) and v = (β,t) T M, we often denote u v := d (u, v), u := d (o , u)= s and ∈ p k − kp p k kp p p u, v := st cos ∢(α, β) = ( u 2 + v 2 u v 2)/2. h ip k kp k kp − k − kp Proposition A.11. Let (M, d) be a geodesic space and p M be fixed. If curv(M) κ for some κ R, ∈ ≥ ∈ then T M is a metric space with curv(T M) 0 (in the sense of Definition A.7). p p ≥ Note that the tangent space is not always a geodesic space, see discussion before Proposition 28 in [Yok12] and the Proposition itself for the proof. Notation . and ., . introduced above is justified by k kp h ip the fact that the cone TpM possesses a Hilbert-like structure described as follows. For a point u = (α, t) and λ 0, we define λ u := (α, λt). Then, the sum of points u, v T M is defined as the mid-point of ≥ · ∈ p 2 u and 2 v as defined in Definition A.3. Finally, it may be checked using the previous definitions that, · · for any u, v T M and any λ 0, we get ∈ p ≥ λ u = λ u and λ u, v = u, λ v = λ u, v . k · kp k kp h · ip h · ip h ip Next we define logarithmic maps. Fix p M. Since M has upper or lower bounded curvature, the ∈ 1 2 angle monotonicity imposes the following observation. If there are two geodesics γx and γx connecting p to x such that ∢(γ1, γ2) = 0, then γ1 = γ2. In other words, the set of points x M for which there is x x x x ∈ not only one equivalence class of geodesics connecting p to x is exactly the set of points x connected to p by at least two distinct geodesics. This set of points is denoted C(p) and called the cut-locus of p. For all x M C(p), we denote x the direction of the unique geodesic γ : [0, 1] M connecting p to x. ∈ \ ↑p x → Definition A.12 (Logarithmic map). Let (M, d) be a geodesic space with curvature bounded from above or below in the sense of Definition A.5 and fix p M. We denote ∈ log : x M C(p) ( x, d(p,x)) T M. p ∈ \ 7→ ↑p ∈ p For all t [0, 1] and all x M C(p), one checks in particular that ∈ ∈ \ log (γ (t)) = t log (x). p x · p More generally, if γ : [0, 1] M is a geodesic in M and if we denote p = γ(t) for some t [0, 1], then → ∈ provided γ(0) = x and γ(1) = y both belong to M C(p), we check that, for all s [0, 1], \ ∈ log (γ(s)) = (1 s) log (x)+ s log (y). p − · p · p Note finally that C(p) is empty for spaces of curvature bounded from above by 0. It is often practical to extend the definition of log to the cut locus C(p). To that aim, it suffices to select, for any x C(p), p ∈ a geodesic connecting p to x. Denoting as above x the direction associated to this chosen geodesic, we ↑p simply extend log to M by setting log (x) = ( x, d(p,x)). This extension depends a priori on the choice of p p ↑p geodesics connecting p to x C(p). However, note that when (M, d) is a Polish space, this extension log ∈ p can be chosen to be measurable on M, by application of the measurable selection theorem of Kuratowski and Ryll-Nardzewski (see Theorem 6.9.3 in [Bog07]). Next is a fundamental result.

Proposition A.13. Let (M, d) be a geodesic space and p M be ﬁxed. If curv(M) 0, then for all ∈ ≥ x,y M, ∈ d(x,y) log (x) log (y) , ≤ k p − p kp with equality if x = p or y = p.

33 Acknowledgements

We would like to thank the associate editor, as well as two anonymous referees, for valuable comments that helped improve signiﬁcantly the original version of the manuscript. We also express our gratitude to Philippe Rigollet for stimulating conversations and useful remarks on a preliminary version of the paper.

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